Properties of the Harmonic and Hyperharmonic Higher Order Gauss Fibonacci Numbers

ABSTRACT In this paper, we define the harmonic higher order Gauss Fibonacci numbers, which are the complex counterparts of higher order Gauss Fibonacci numbers and have been previously studied in the literature. Due to the absence of a known closed form for harmonic Fibonacci numbers, we also generalize their complex counterparts with respect to a certain parameter . The computational analysis in this case is significantly more complex and intricate. Furthermore, for varying values of , we can derive several sequences of harmonic complex Fibonacci‐type numbers. Using the difference operator and its properties, we give formulas for these numbers. Given their complexity, we present examples using Maple to verify our results. We define hyperharmonic higher order Gauss Fibonacci numbers, encompassing harmonic higher order Gauss Fibonacci numbers. Lastly, we give generating functions for harmonic and hyperharmonic higher order Gauss Fibonacci numbers via the generalized ‐logarithm function.